# De Morgan's Laws (Set Theory)/Relative Complement

## Theorem

Let $S, T_1, T_2$ be sets such that $T_1, T_2$ are both subsets of $S$.

Then, using the notation of the relative complement:

### Complement of Intersection

$\relcomp S {T_1 \cap T_2} = \relcomp S {T_1} \cup \relcomp S {T_2}$

### Complement of Union

$\relcomp S {T_1 \cup T_2} = \relcomp S {T_1} \cap \relcomp S {T_2}$

### General Case

Let $S$ be a set.

Let $T$ be a subset of $S$.

Let $\mathcal P \left({T}\right)$ be the power set of $T$.

Let $\mathbb T \subseteq \mathcal P \left({T}\right)$.

Then:

#### Complement of Intersection

$\displaystyle \relcomp S {\bigcap \mathbb T} = \bigcup_{H \mathop \in \mathbb T} \relcomp S H$

#### Complement of Union

$\displaystyle \complement_S \left({\bigcup \mathbb T}\right) = \bigcap_{H \mathop \in \mathbb T} \complement_S \left({H}\right)$

### Family of Sets

Let $S$ be a set.

Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$.

Then:

#### Complement of Intersection

$\displaystyle \relcomp S {\bigcap_{i \mathop \in I} \mathbb S_i} = \bigcup_{i \mathop \in I} \relcomp S {S_i}$

#### Complement of Union

$\displaystyle \relcomp S {\bigcup_{i \mathop \in I} \mathbb S_i} = \bigcap_{i \mathop \in I} \relcomp S {S_i}$

## Source of Name

This entry was named for Augustus De Morgan.